A monomial ideal <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$I\subseteq \mathbb {K}[x_1,\ldots , x_n]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo>⊆</mo> <mi mathvariant="double-struck">K</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> is called a Simis ideal if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$I^{(s)}=I^s$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>I</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>I</mi> <mi>s</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$s\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$I^{(s)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>I</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> denotes the s-th symbolic power of I. Let I be a support-2 monomial ideal such that its irreducible primary decomposition is minimal. We prove that I is a Simis ideal if and only if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sqrt{I}$</EquationSource> <EquationSource Format="MATHML"><math> <msqrt> <mi>I</mi> </msqrt> </math></EquationSource> </InlineEquation> is Simis and I has a standard linear weighting. This result thereby proves a recent conjecture for the class of support-2 monomial ideals proposed by Mendez, Pinto, and Villarreal. Furthermore, we give a complete characterization of the Cohen-Macaulay property for support-2 monomial ideals whose radical is the edge ideal of a whiskered graph. Finally, we classify when these ideals are Simis in degree 2.

Support-2 Monomial Ideals that are Simis

期刊论文
发表日期：

2026年4月28日

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Paromita Bordoloi,
Kanoy Kumar Das,
Rajiv Kumar

摘要

A monomial ideal $I\subseteq \mathbb {K}[x_1,\ldots , x_n]$ $I \subseteq K [x_{1}, \dots, x_{n}]$ is called a Simis ideal if $I^{(s)}=I^s$ $I^{(s)} = I^{s}$ for all $s\ge 1$ $s \geq 1$ , where $I^{(s)}$ $I^{(s)}$ denotes the s-th symbolic power of I. Let I be a support-2 monomial ideal such that its irreducible primary decomposition is minimal. We prove that I is a Simis ideal if and only if $\sqrt{I}$ $\sqrt{I}$ is Simis and I has a standard linear weighting. This result thereby proves a recent conjecture for the class of support-2 monomial ideals proposed by Mendez, Pinto, and Villarreal. Furthermore, we give a complete characterization of the Cohen-Macaulay property for support-2 monomial ideals whose radical is the edge ideal of a whiskered graph. Finally, we classify when these ideals are Simis in degree 2.